Theory and Problems of Digital Principles

Theory and Problems of Digital Principles

Digital electronics is a rapidly growing technology. Digital circuits are used in
most new consumer products, industrial equipment and controls, and office,
medical, military, and communications equipment. This expanding use of digital
circuits is the result of the development of inexpensive integrated circuits and the
application of display, memory, and computer technology.

ROGER L. TOKHEIM holds B.S., M.S., and Ed.S. degrees from
St. Cloud State University and the University of Wisconsin-Stout. He is
the author of Digital Electronics and its companion Activities Manual for
Digital Electronics, Schaum ’s Outline of Microprocessor Fundamentals, and
numerous other instructional materials on science and technology. An
experienced educator at the secondary and college levels, he is presently
an instructor of Technology Education and Computer Science at Henry
Sibley High School, Mendota Heights, Minnesota.
Schaum’s Outline of Theory and Problems of
DIGITAL PRINCIPLES
Copyright 0 1994, 1988, 1980 by The McGraw-Hill Companies, Inc. All Rights Reserved. Printed
in the United States of America. Except as permitted under the Copyright Act of 1976, no part of
this publication may be reproduced or distributed in any form or by any means, or stored in a data
base or retrieval system. without the prior written permission of the publisher.
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ISBN 0-07-0b5050-0
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Library of Congress Cataloging-in-Publication Data
Tokheim, Roger L.
Schaum’s outline of theory and problems of digital prinicples/by
Roger L. Tokheim-3rd ed.
p. cm.-(Schaum’s outline series)
Includes index.
ISBN 0-07-065050-0
1. Digital electronics. I. ‘Title. 11. Series.
TK7868.D5T66 1994 93-64
62 1.3815-dc20 CIP
McGraw -Hill
A Division of The McGraw-HiUCompanies
z

Digital electronics is a rapidly growing technology. Digital circuits are used in
most new consumer products, industrial equipment and controls, and office,
medical, military, and communications equipment. This expanding use of digital
circuits is the result of the development of inexpensive integrated circuits and the
application of display, memory, and computer technology.
Schaum’s Outline of Digitd Principles provides inforrnation necessary to lead
the reader through the solution of those problems in digital electronics one might
encounter as a student, technician, engineer, or hobbyist. While the principles of
the subject are necessary, the Schaum’s Outline philosophy is dedicated to showing
the student how to apply the principles of digital electronics through practical
solved problems. This new edition now contains over 1000 solved and supplemen-
tary problems.
The third edition of Schaum’s Outline of Digital Principles contains many of
the same topics which made the first two editions great successes. Slight changes
have been made in many of the traditional topics to reflect the technological trend
toward using more CMOS, NMOS, and PMOS integrated circuits. Several micro-
processor/microcomputer-related topics have been included, reflecting the current
practice of teaching a microprocessor course after or with digital electronics. A
chapter detailing the characteristics of TTL and CMOS devices along with several
interfacing topics has been added. Other display technologies such as liquid-crystal
displays (LCDs) and vacuum fluorescent (VF) displays have been given expanded
coverage. The chapter on microcomputer memory has been revised with added
coverage of hard and optical disks. Sections on programmable logic arrays (PLA),
magnitude comparators, demultiplexers, and Schmitt trigger devices have been
added.
The topics outlined in this book were carefully selected to coincide with
courses taught at the upper high school, vocational-Iechnical school, technical
college, and beginning collcge level. Several of the most widely used textbooks in
digital electronics were analyzed. The topics and problems included in this
Schaum’s Outline reflect those encountered in standard textbooks.
Schuiini’s Outline of Digital Principles, Third Edition, begins with number
systems and digital codes and continues with logic gates and combinational logic
circuits. It then details the characteristics of both TTL and CMOS ICs, along with
various interfacing topics. Next encoders, decoders, and display drivers are ex-
plored, along with LED, LCD, and VF seven-segment displays. Various arithmetic
circuits are examined. It then covers flip-flops, other rnultivibrators, and sequential
logic, followed by counters and shift registers. Next semiconductor and bulk
storage memories are explored. Finally, niul tiplexers, demultiplexers, latches and
buffers, digital data transmission, magnitude comparators, Schmitt trigger devices,
and programmable logic arrays are investigated. The book stresses the use of
industry-standard digital ICs (both TTL and CMOS) so that the reader becomes
familiar with the practical hardware aspects of digital electronics. Most circuits in
this Schaum’s Outline can be wired using standard digital ICs.
I wish to thank my son Marshall for his many hours of typing, proofreading,
and testing circuits to make this book as accurate as possible. Finally, I extend my
appreciation to other family members Daniel and Carrie for their help and
patience.
ROGER TOKHEIM
L.
...
111

Chapter 1
Numbers Used in Digital Electronics
1-1 INTRODUCTION
The decimal number system is familiar to everyone. This system uses the symbols 0, 1, 2, 3, 4, 5, 6,
7, 8, and 9. The decimal system also has a place-value characteristic. Consider the decimal number
238. The 8 is in the 1s position or place. The 3 is in the 10s position, and therefore the three 10s stand
for 30 units. The 2 is in the 100s position and means two loos, or 2,OO units. Adding 200 + 30 + 8 gives
the total decimal number of 238. The decimal number system is also called the base 10 system. It is
referred to as base 10 because it has 10 different symbols. The base 10 system is also said to have a
radix of 10. “Radix” and “base” are terms that mean exactly the same thing.
Binary numbers (base 2) are used extensively in digital electronics and computers. Both hexadeci-
mal (base 16) and octal (base 8) numbers are used to represent groups of binary digits. Binary and
hexadecimal numbers find wide use in modern microcomputers.
All the number systems mentioned (decimal, binary, octal, and hexadecimal) can be used for
counting. All these number systems also have the place-value cha.racteristic.
1-2 BINARY NUMBERS
The binary number system uses only two symbols (0,l). It is said to have a radix of 2 and is
commonly called the base 2 number system. Each binary digit is called a bit.
Counting in binary is illustrated in Fig. 1-1. The binary number is shown on the right with its
decimal equivalent. Notice that the least significant bit (LSB) is the 1s place. In other words, if a 1
appears in the right column, a 1 is added to the binary count. The second place over from the right is
the 2s place. A 1 appearing in this column (as in decimal 2 row> means that 2 is added to the count.
Three other binary place values also are shown in Fig. 1-1 (4s, 8s, and 16s places). Note that each
larger place value is an added power of 2. The 1s place is really Z0, the 2s place 2*,the 4s place 22, the
8s place 23, and the 16s place z4. It is customary in digital electronics to memorize at least the binary
counting sequence from 0000 to 1111 (say: one, one, one, one) or decimal 15.
Consider the number shown in Fig. 1-2a. This figure shows how to convert the binary 10011 (say:
one, zero, zero, one, one) to its decimal equivalent. Note that, for each 1 bit in the binary number, the
decimal equivalent for that place value is written below. The decimal numbers are then added
(16 + 2 + 1 = 19) to yield the decimal equivalent. Binary 10011 then equals a decimal 19.
Consider the binary number 101110 in Fig. 1-2b. Using the same procedure, each 1 bit in the
binary number generates a decimal equivalent for that place value. The most signijicant bit (MSB) of
the binary number is equal to 32. Add 8 plus 4 plus 2 to the 32 for a total of 46. Binary 101110 then
equals decimal 46. Figure 1-2b also identifies the binary point (similar to the decimal point in decimal
numbers). It is customary to omit the binary point when working with whole binary numbers.
What is the value of the number 111? It could be one hundred and eleven in decimal or one, one,
one in binary. Some books use the system shown in Fig. 1-2c to designate the base, or radix, of a
number. In this case 10011 is a base 2 number as shown by the small subscript 2 after the number. The
number 19 is a base 10 number as shown by the subscript I0 after the number. Figure 1-2c is a
summary of the binary-to-decimal conversions in Fig. 1-2a and b.
How about converting fractional numbers? Figure 1-3 illustrates the binary number 1110.101
being converted to its decimal equivalent. The place values are given across the top. Note the value of
each position to the right of the binary point. The procedure for making the conversion is the same as
with whole numbers. The place value of each 1 bit in the binary number is added to form the decimal
number. In this problem 8 + 4 + 2 + 0.5 + 0.125 = 14.625 in decimal.
I

CHAP. 11 NUMBERS USED IN DIGITAL ELECTRONICS 3
1 1
~ ~ ~
Powers o f 2 23 22 2' 2O 1!2' 1/z2 1p3
--
4s 2s Is 0.5s 0.25s 0.125s
~~~~~~~~
Binary 1 1 1 0 . 1 0 1
Decimal 8 + 4 + 2 + 0.5 + 0.125 = 14.625
Fig. 1-3 Binary-to-decimal conversion
Convert the decimal number 87 to a binary number. Figure 1-4 shows a convenient method for
making this conversion. The decimal number 87 is first divided by 2, leaving 43 with a remainder of 1.
The remainder is important and is recorded at the right. It becomes the LSB in the binary number.
The quotient (43) then is transferred as shown by the arrow and becomes the dividend. The quotients
are repeatedly divided by 2 until the quotient becomes 0 with a remainder of 1, as in the last line of
Fig. 1-4. Near the bottom the figure shows that decimal 87 equals binary 1010111.
87; -+ 2 = 4" remainder of 1 I.SB
43 i 2 = f l remainder of 1 1
+7
21 -+ 2 = 10 remainder
10 -+ 2 =
of 1
remainder of 0I--
4
5 -+ 2 = 2 remainder of 1
F---l remainder of 0
2 i2 = 1 1
1
-1
5----1remainder of 1
1 +2 0 =
87,,-l
1
MSH III
0 1 01 1
"
1 1,
Fig. 1-4 Decimal-to-binary conversion
Convert the decimal number 0.375 to a binary number. Figure 1-51 illustrates one method of
performing this task. Note that the decimal number (0.375) is being multiplied by 2. This leaves a
product of 0.75. The 0 from the integer place (1s place) becomes the bit nearest the binary point. The
0.75 is then multiplied by 2, yielding 1.50. The carry of 1 to the integer (1s place) is the next bit in the
binary number. The 0.50 is then multiplied by 2, yielding a product of 1.00. The carry of 1 in the
integer place is the final 1 in the binary number. When the product is 1.00, the conversion process is
complete. Figure 1-5a shows a decimal 0.375 being converted into a binary equivalent of 0.011.
Figure 1-5b shows the decimal number 0.84375 being Converted into binary. Again note that
0.84375 is multiplied by 2. The integer of each product is placed below, forming the binary number.
When the product reaches 1.00, the conversion is complete. This problem shows a decimal 0.84375
being converted to binary 0.1 1011.
Consider the decimal number 5.625. Converting this number to binary involves two processes. The
integer part of the number ( 5 ) is processed by repeated division near the top in Fig. 1-6. Decimal 5 is
converted to a binary 101. The fractional part of the decimal number (.625) is converted to binary .101
at the bottom in Fig. 1-6. The fractional part is converted to binary through the repeated multiplication
process. The integer and fractional sections are then combined to show that decimal 5.625 equals
binary 101.101.

8 NUMBERS USED IN DIGITAL ELECTRONICS [CHAP. 1
hexadecimal number. The process is complete because the integer part of the quotient is 0. The
process in Fig. 1-9a converts the decimal number 45 to the hexadecimal number 2D.
Convert the decimal number 250.25 to a hexadecimal number. The conversion must be done by
using two processes as shown in Fig. 1-9b. The integer part of the decimal number (250) is converted
to hexadecimal by using the repeated divide-by-16 process. The remainders of 10 (A in hexadecimal)
and 15 (F in hexadecimal) form the hexadecimal whole number FA. The fractional part of the 250.25
is multiplied by 16 (0.25 X 16). The result is 4.00. The integer 4 is transferred to the position shown in
Fig. 1-9b. The completed conversion shows the decimal number 250.25 equaling the hexadecimal
number FA.4.
The prime advantage of the hexadecimal system is its easy conversion to binary. Figure 1-10a
shows the hexadecimal number 3B9 being converted to binary. Note that each hexadecimal digit forms
a group of four binary digits, or bits. The groups of bits are then combined to form the binary number.
In this case 3B9,, equals 1110111001,.
( a ) Hexadecimal-to-binaryconversion
(6) Fractional hexadecimal-to-binaryconversion
1010 1000 0101 0
1 1 1 101010000101~ A8516
=
A 8 5
(c) Binary-to-hexadecimalconversion
( d ) Fractional binary-to-hexadecimalconversion
Fig. 1-10
Another hexadecimal-to-binary conversion is detailed in Fig. 1-106. Again each hexadecimal digit
forms a 4-bit group in the binary number. The hexadecimal point is dropped straight down to form the
binary point. The hexadecimal number 47.FE is converted to the binary number 1000111.1111111.It is
apparent that hexadecimal numbers, because of their compactness, are much easier to write down
than the long strings of 1s and OS in binary. The hexadecimal system can be thought of as a shorthand
method of writing binary numbers.
Figure 1-1Oc shows the binary number 101010000101being converted to hexadecimal. First divide
the binary number into 4-bit groups starting at the binary point. Each group of four bits is then
translated into an equivalent hexadecimal digit. Figure 1-10c shows that binary 101010000101equals
hexadecimal A85.
Another binary-to-hexadecimal conversion is illustrated in Fig. 1-10d. Here binary 10010.011011 is
to be translated into hexadecimal. First the binary number is divided into groups of four bits, starting
at the binary point. Three OS are added in the leftmost group, forming 0001. Two OS are added to the
rightmost group, forming 1100. Each group now has 4 bits and is translated into a hexadecimal digit as
shown in Fig. 1-10d. The binary number 10010.011011 then equals 12.6C,,.
As a practical matter, many modern hand-held calculators perform number base conversions.
Most can convert between decimal, hexadecimal, octal, and binary. These calculators can also perform
arithmetic operations in various bases (such as hexadecimal).

10 NUMBERS USED IN DIGITAL ELECTRONICS [CHAP. 1
Solution:
Follow the procedure shown in Fig. 1-10a and b. Refer also to Fig. 1-7. The binary equivalents of the
hexadecimal numbers are as follows:
(U) B,, = 1011, (c) lC,, = 11100, ( e ) lF.C,, = 11111.11,
( b ) E,, = 1110, ( d ) A64,, = 101001100100, ( f ) 239.4,, = 1000111001.01,
1.22 Convert the following binary numbers to their hexadecimal equivalents:
( a ) 1001.1111 ( c ) 110101.O11001 ( e ) IOIOOI 11.I11011
( b ) 10000001.1101 ( d ) 10000.1 ( f ) 1000000.0000111
Solution:
Follow the procedure shown in Fig. 1-1Oc and d . Refer also to Fig. 1-7. The hexadecimal equivalents
of the binary numbers are as follows:
(U) 1001.11112 = 9.F1, (c) 110101.011001, = 35-64,, ( e ) 10100111.111011, = A’I.EC,,
( b ) 10000001.1101, = 81.D,, ( d ) 10000.1, = 10.8,, ( J ’ ) 1000000.0000111, = 40.0E1,
1-4 2s COMPLEMENT NUMBERS
The 2s complement method of representing numbers is widely used in microprocessor-based
equipment. Until now, we have assumed that all numbers are positive. However, microprocessors must
process both positive and negative numbers. By using 2s complement representation, the sign as well as
the magnitude of a number can be determined.
Assume a microprocessor register 8 bits wide such as that shown in Fig. 1-llu. The most-
significant bit (MSB) is the sign bit. If this bit is 0, then the number is ( + ) positive. However, if the
sign bit is 1, then the number is ( - ) negative. The other 7 bits in this 8-bit register represent
the magnitude of the number.
The table in Fig. 1-llb shows the 2s complement representations for some positive and negative
numbers. For instance, a + 127 is represented by the 2s complement number 01111111. A decimal
- 128 is represented by the 2s complement number 10000000. Note that the 2s complement representa-
tions for allpositiiie ualues are the same as the binary equivalents for that decimal number.
Convert the signed decimal -1 to a 2s complement number. Follow Fig. 1-12 as you make the
conversion in the next five steps.
Step 1. Separate the sign and magnitude part of - 1. The negative sign means the sign bit will be
1 in the 2s complement representation.
Step 2. Convert decimal 1 to its 7-bit binary equivalent. In this example decimal 1 equals
0000001 in binary.
Step 3. Convert binary 0000001 to its Is complement form. In this example binary 0000001
equals 1111110 in Is complement. Note that each 0 is changed to a 1 and each 1 to a 0.
Step 4. Convert the 1s complement to its 2s complement form. In this example 1s complement
1111110 equals 1111111 in 2s complement. Add + 1 to the 1s complement to get the 2s
complement number.
Step 5. The 7-bit 2s complement number (1111111 in this example) becomes the magnitude part
of the entire 8-bit 2s complement number.
The result is that the signed decimal - 1 equals 11111111 in 2s complement notation. The 2s
complement number is shown in the register near the top of Fig. 1-12.